Proof that a r.v. with a particular random walk is a martingale











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Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that



$P(X_1=1)=1-P(X_1=-1)=p$



$p in (0, frac{1}{2})$



Consider a random walk $S_n=sum_{i=0}^n X_i$



Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that



$tau = min [n in mathbb{N}:S_n=0]$



Define



$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$



I need to show that $(W_n,mathcal{F}_n)$ is a martingale



Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.










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  • $W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
    – Kavi Rama Murthy
    Nov 22 at 10:11










  • Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
    – Kavi Rama Murthy
    Nov 22 at 10:14















up vote
2
down vote

favorite












Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that



$P(X_1=1)=1-P(X_1=-1)=p$



$p in (0, frac{1}{2})$



Consider a random walk $S_n=sum_{i=0}^n X_i$



Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that



$tau = min [n in mathbb{N}:S_n=0]$



Define



$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$



I need to show that $(W_n,mathcal{F}_n)$ is a martingale



Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.










share|cite|improve this question









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user618532 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • $W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
    – Kavi Rama Murthy
    Nov 22 at 10:11










  • Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
    – Kavi Rama Murthy
    Nov 22 at 10:14













up vote
2
down vote

favorite









up vote
2
down vote

favorite











Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that



$P(X_1=1)=1-P(X_1=-1)=p$



$p in (0, frac{1}{2})$



Consider a random walk $S_n=sum_{i=0}^n X_i$



Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that



$tau = min [n in mathbb{N}:S_n=0]$



Define



$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$



I need to show that $(W_n,mathcal{F}_n)$ is a martingale



Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.










share|cite|improve this question









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user618532 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that



$P(X_1=1)=1-P(X_1=-1)=p$



$p in (0, frac{1}{2})$



Consider a random walk $S_n=sum_{i=0}^n X_i$



Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that



$tau = min [n in mathbb{N}:S_n=0]$



Define



$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$



I need to show that $(W_n,mathcal{F}_n)$ is a martingale



Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.







probability martingales random-walk






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edited Nov 22 at 10:50









Bernard

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asked Nov 22 at 10:08









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user618532 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • $W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
    – Kavi Rama Murthy
    Nov 22 at 10:11










  • Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
    – Kavi Rama Murthy
    Nov 22 at 10:14


















  • $W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
    – Kavi Rama Murthy
    Nov 22 at 10:11










  • Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
    – Kavi Rama Murthy
    Nov 22 at 10:14
















$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11




$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11












Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14




Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14















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